Long-RunRiskandHiddenGrowthPersistence Pakos,Michal MunichPersonalRePEcArchive

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Long-Run Risk and Hidden Growth Persistence

Pakos, Michal

Michal Pakos

17 April 2013

Online at https://mpra.ub.uni-muenchen.de/47217/

MPRA Paper No. 47217, posted 17 Jul 2013 08:38 UTC

Long-Run Risk and Hidden Growth Persistence

Michal Pakoš¹

CERGE-EI FEBRUARY 2013

1CERGE-EI, a joint workplace of Charles University in Prague and the Economics Institute of the Academy of Sciences of the Czech Republic, Politických Vězňů 7, 111 21 Prague 1, Czech Republic. Email: michal.pakos@cerge-ei.cz. URL:home.cerge-ei.cz/pakos. Financial support of the Czech Science Foundation(grant no. P403/11/2288) is gratefully acknowledged. Helpful comments from Byeongju Jeong, John Campbell, John Cochrane, George Constantinides, Max Gillman, Rick Green, Lars Hansen, John Heaton, Jan Hanousek, Burton Hollifield, Štepán Jurajda, Michal Kejak, Jan Kmenta, Christopher Otrok, Luboš Pástor, Bryan Routledge, Sergey Slobodyan, Chris Telmer, Pietro Veronesi, Evangelia Vourvachaki, Amir Yaron, Stan Zin, and Petr Zemčík, the participants of the Netherlands Econometric Study Group (NESG) Annual Meeting (Leuven, June 2010), the discussant (Ivan Shaliastovich) at the Western Finance Association Meeting 2011 (Santa Fe, New Mexico), and the participants of the 2012 Meetings of the Society for Computational Economics 2012 (Prague, Czech Republic) as well as the 2012 European Meeting of the Econometric Society (Málaga, Spain). The paper previously circulated under the name "Consumption, Asset Prices and Persistent Economic Uncertainty".

Abstract

An extensive literature has analyzed the implications of hidden shifts in the dividend growthrate. However, corresponding research on learning about growth persistence is completely lacking. Hidden persistence is a novel way to introduce long-run risk into standard business-cycle models of asset prices because it tightly intertwines the cyclical and long-run frequencies. Hidden per- sistence magnifies endogenous changes in the forecast variance of the long-run dividend growth rate despite homoscedastic consumption innovations. Not only does changing forecast variance make discrimination between protracted spells of anemic growth and brief business recessions difficult, it also endogenously induces additional variation in asset price discounts due to the preference for early uncertainty resolution.

Keywords: Asset Pricing, Learning, Hidden Persistence, Forecast Variance, Eco- nomic Uncertainty, Business Cycles, Long-Run Risk, Peso Problem, Timing Premium

JEL:E13, E21, E27, E32, E37, E44, G12, G14

1 Introduction

Asset prices contain both cyclical as well as long-run components. The cyclical com- ponent has been documented empirically byFama and French(1988a,b,1989,1990), Ferson and Harvey (1991), Campbell and Diebold (2009), Backus, Routledge, and Zin (2010), Lustig and Verdelhan (2013) and many others. The long-run-risk com- ponent has been shown to arises from either consumption smoothing as inLochstoer and Kaltenbrunner (2010) or learning about the mean growth rate as in ?. In re- sponse, two separate strands within the finance literature have developed. The first one accentuates solely the cyclical component in the expected dividends and dis- count rates as the key driver of asset prices. Important theoretical contributions that model the cyclical component as a Markov chain include Abel (1994),Brandt, Zeng, and Zhang (2004), Cagetti et al. (2002), Cecchetti, Lam, and Mark (1990, 1993,2000a),Ju and Miao(2012) , Kandel and Stambaugh(1990),Lettau, Ludvig- son, and Wachter (2008) andVeronesi(2004). Such important literature however is only partially successful in accounting for the salient features of asset prices.

The second strand of the asset pricing literature accentuates the long-run com- ponent. Bansal and Yaron (2004); Bansal, Gallant, and Tauchen (2007); Bansal and Shaliastovich (2010), Hansen, Heaton, and Li (2008) and Parker and Julliard (2005) are the key contributions. The long-run risk literature is significantly more successful in accounting for a variety of observed asset-pricing phenomena. However, the exclusive reliance on the long-run risk component suggests that the variation in expected business conditions is irrelevant. In fact, Bansal, Kiku, and Yaron (2010) argue that the cyclical risk premiums are negligible and as a result changing forecasts of business conditions do not matter for asset pricing. ¹ It seems however that such

1In fact, the cyclical risk premiums are significant only for low values of the inter temporal elasticity of substitution which leads to counter-factually high and volatile risk-free rates.

conclusion directly contradicts the extensive empirical literature that accentuates the cyclical component in asset prices.

The main thesis of our paper is that expected business conditions matter for asset pricing even in long-run risk models. We build upon the important research agenda of Bansal, Kiku, and Yaron (2010) and model consumption and dividend growth rate dynamics at two frequencies jointly: the cyclical one and the long-run one. We do not posit complete information as Bansal et al., however. In fact, we assume that perfect discrimination among the two components is not feasible. We model such incomplete information in the manner that has become standard in the modern business-cycle literature: subject the consumption and dividends to hidden shifts across various phases of the business cycle. This is consistent withBurns and Mitchell(1946) who emphasize the different dynamics of economic aggregates across the various phases of the business cycle.

Subjecting growth persistence to hidden Markov shifts is novel but could not be more relevant. First, vast literature has analyzed the implications of hidden Markov shifts only in the rate of the growth. Considering persistence is a natural extension.

For example, AR(1) process has two key parameters: the long-run mean to which the process tends to revert but also the persistence, the speed of the adjustment toward the mean.

Second, stochastic persistence introduces long-run components into asset prices in a very natural manner. Moreover, the speed of adjustment in the AR(1) process for the expected consumption growth in Bansal and Yaron (2004) suddenly has a structural interpretation in terms of the average duration of high and low growth periods.

Third, hidden growth persistence nests both the business-cycle and long-run risk models of asset prices within a single framework, thereby allowing to evaluate relative

contributions of each component to risk premiums.

Fourth, the fact that the persistence is actually hidden intertwines the cyclical and the long-run components, effectively bringing the long-run asset-price dynamics to the business-cycle frequency. The reason is that the inference problem with respect to the hidden duration of the recession or lost decade gives rise to “Peso” problem, a situation where even a small probability that the growth might be protracted affects asset prices dramatically. ²

Fifth, hidden persistence dramatically magnifies the fluctuations in the forecast variance of the long-run dividend growth. Learning about persistence is thus an alternative way, in comparison to the exogenously fluctuating consumption volatility in Bansal and Yaron, to induce persistent variation in economic uncertainty and thus to generate endogenous changes in the conditional distribution of asset prices.

Sixth, the relevance of modeling growth persistence is also empirical and stems from a simple back-of-the-envelope calculation showing that decade-long spells of anemic growth in commonly calibrated two-state models with constant transition probability matrices occur about every 22,000th recession which is far too rare. ³

The virtue of our modeling approach is that the hidden persistence is naturally nested within the Hamilton (1989) hidden Markov chain setting when a subset of hidden states have exactly the same growth rate and just differ in their average

2The Peso problem refers to a situation when the possibility of some infrequent event, such as significantly higher persistence of low growth, has an effect on asset prices. The event may or may not have even happened in the sample but it must be difficult to accurately predict using economic history. Note that learning about the hidden persistence of economic growth fits this description well. SeeEvans(1996) for a review of the Peso literature.

3Assume that the recession has mean duration of about a year. That is consistent with the point estimateλ22 presented in Table2. Then, the probability of randomly drawing an economic slowdown of length 10 years or more equals

P(T >10|recession) =ˆ ∞ 10

e^−tdt≈0.00005.

Thus, it takes about1/0.00005≈22,000full recessions for the lost decade to arise. That however is far too rare for it to matter for asset pricing. In our calibration, lost decades occur once a century on average.

duration. Such stochastic duration, or persistence, may be economically thought of as the outcome of several, in our case two, distinct types of economic slowdowns, for example. The first type includes business-cycle slowdowns: the fairly common but brief fluctuations in economic activity that tend to last on average about 3 to 5 quarters. The second type may be thought of as “lost decades”, that is, economic slowdowns that are less likely to occur but tend to be very protracted (Japan’s lost decade is a prime example).

Although our underlying Markov chain technically features three rather than two states, the growth rate shifts between high-growth and low-growth only. That happens because the third state is needed solely to model the slowdown persistence.

In fact, as inTauchen(1986a,b) andTauchen and Hussey(1991), our Markov chain may be thought of as a discretized AR(1) process where the persistence parameter is in addition hidden. Statistically speaking, the probability distribution of the recession duration has heavier tails than the usual exponential distribution that arises in continuous-time Markov chains with constant transition probability intensities.

The fact that the second and third states feature exactly the same growth rate but just one tends to last so much longer is substantially different from the Markov- chain setting inRietz(1988),Barro(2006) andJohannes, Lochstoer, and Mou(2012) where the extension of the two-state chain of Mehra and Prescott (1985b) to three states is dictated by the need to model quite a rare disastrous growth rate, one in which the economy shrinks at a large negative rate well below the rate observed during recessions. Moreover, except for Johannes et al., these studies do not focus on learning as their driving Markov model is in fact observable.

We follow the long-run risk literature and endow the representative investor with the preferences of Kreps and Porteus (1978) and Epstein and Zin (1989, 1991) as extended to the continuous-time setting byDuffie and Epstein(1992). Thus, our in-

vestor cares deeply about the timing profile of the resolution of economic uncertainty.

Relaxing the independence axiom of the expected utility framework is relevant as Dreeze and Modigliani (1972) observe that the utility in dynamic models is derived not only from the level of state-contingent consumption over time but also from the way in which the uncertainty about future resolves over time. Such timing of the resolution of uncertainty entails no utility gain or loss in a framework built around expected utility theory of von Neuman and Morgenstern. In the words ofChew and Epstein(1989) andMa(1998), investors do not demand extra risk premiums or price discounts for entering actuarially fair gambles the uncertainty of which resolves far in the future rather than now. Thus, the so-called “timing premiums in expected returns” are absent. In our setting, however, the significant and persistent fluctua- tions in the investor’s beliefs regarding future rate as well as persistence of economic growth do induce corresponding variation and persistence of the forecast variance of the long-run dividend growth and thus generate additional “timing discounts” in asset prices. The setting of Markov shifts dramatically improves the match of the salient features of the conditional distributions of asset prices when the shifts are hidden rather than observable as inAbel (1994).

There are in addition a few important related papers. Cecchetti, Lam, and Mark (2000b) study near-rational investors who tend to underestimate the average duration of economic slowdowns. Shorter slowdowns however tend to make the risk premiums for cyclical fluctuations in Bansal, Kiku, and Yaron (2010) lower. Next, Johannes, Lochstoer, and Mou(2012) explore Bayesian model averaging of the two- state ( à laMehra and Prescott(1985a)) and three-state (à laRietz(1988) andBarro (2006)) hidden Markov chain for consumption growth. Their model does not feature a long-run dividend risk. Finally, Ai (2010) and Edgea, Laubacha, and Williams (2007) study learning about the long-run mean of AR(1) process. However, Kalman

filter does not allow to learn about the persistence of AR(1) process.

The paper is organized as follows. We present the formal model and derive the theoretical asset pricing implications of hidden growth persistence in Section 2.

We describe data in Section 3 and the results of statistical inference in Section 4.

We describe the empirical asset pricing implications of hidden growth persistence in Section 5. We conclude in Section 6. The detailed mathematical proofs are relegated to online Appendix.

2 Model

We start the analysis by describing the investor’s preferences. After that, we specify the dynamics of the dividends from all the assets. We then solve the recursive Bayesian inference problem. Next, we decompose the total variance of the long-run dividend growth rate into the variance of the conditional mean plus the mean of the conditional variance, show that the former changes endogenously whereas the latter is a constant and interpret economic uncertainty in terms of the variability of the mean long-run forecasts. Finally, we set up consumption-portfolio problem and using the first-order conditions we find the equilibrium asset prices.

2.1 Preferences

Consider an endowment economy with a single long-lived asset which yields com- pletely perishable consumption good. Furthermore, endow the representative in- vestor with the recursive utility

v_t = E_t ˆ _∞

t

u(c_τ, v_τ) dτ

,

defined over the perishable consumption rate stream ct and the continuation utility vt. ⁴ The normalized inter-temporal utility aggregator uover the consumption rate and the continuation utility is

u(c, v) = δ 1−_ψ¹

c^1−ψ1 −((1−γ)v)

1−1 ψ 1−γ

((1−γ)v)

1−1 ψ 1−γ−1

. (2.1)

The parameter δ is the subjective discount rate, the parameter γ is the coefficient of the relative risk aversion in terms of atemporal and actuarially-fair gambles over the wealth, and the parameter ψ is the elasticity of inter-temporal substitution.

The preference for early resolution of uncertainty is reflected in the convexity of the inter temporal aggregatoru(c, v)with respect to the second argument; for example, agents prefer early resolution of uncertainty for γ > _ψ¹. The expected utility of von Neuman and Morgenstern is nested as a special case ofγ = _ψ¹.

2.2 Asset Markets and Dividends

We consider two asset classes: equities and bonds. For equities, we in addition distinguish between unlevered and levered equity. Unlevered equity, sometimes also called consumption claim, corresponds to the standard Lucas tree with which the representative agent is endowed. It is the only asset in positive net supply and we normalize it to one. The levered equity corresponds to the aggregate equity market and it may be thought of as a levered consumption claim. For bonds, we consider only purely discount real bonds that pay zero coupons, D^b_t = 0. We denote the universe of assets A = {u, l, b} where u is the unlevered equity, l is levered equity and bis a purely discount real bond.

4We agree that modeling consumer durable goods and housing is worthwhile but unfortunately it augments the state space model in the dynamic programming exercise. See in particular Aït- Sahalia, Parker, and Yogo(2004), Pakoš(2006, 2011), Piazzesi, Schneider, and Tuzel(2007) and Yogo(2006).

We specify the dynamics of the dividend streams D^u_t and D^l_t from the equities as a bivariate hidden Markov model in logs:

d logD_t^u = µ^u_Stdt+σ^udZ_t^u, (2.2) d logD^l_t = µ^l_Stdt+σ^ldZ_t^l, (2.3)

where Z_t^u and Z_t^l are two uncorrelated Brownian Motions. The model is an exten- sion of Hamilton (1989) and Cecchetti, Lam, and Mark (1993). The instantaneous dividend volatilitiesσ^u andσ^l are constant whereas the predictable components µ^e_S

t

for e∈ E ={u, l} are driven by the common Markov chain St with the state space

S={1 =expansion,2 =recession,3 =lost decade}.

The unobservability of the underlying state induces endogenously time-varying un- certainty due to inference problems. All dividend parameters are estimated by max- imum likelihood from the postwar U.S. consumption and dividend data.

2.2.1 Expected Endowment Rate

Our specification of the hidden Markov chain differs from the outstanding literature.

Standard models of business-cycle fluctuations in the economic activity are naturally modeled as a two-state Markov chain with the state space {1 = high growth,2 = low growth}. The high-growth state corresponds to expansions and the low-growth state to recessions. ⁵ Furthermore, the transition probability matrix is typically assumed constant. In our setting of very brief decision interval h, such assumption

5Lettau, Ludvigson, and Wachter(2008) in their explorations of the role of the Great Modera- tion in understanding the equilibrium asset prices also consider independent regime switches in the consumption volatility. As we shall see hereafter, the volatility of thetime−aggregatedconsump- tion growth rate does change in our setting endogenously due to the hidden regime shifts and we do not need to model the variation in consumption volatility as an additional Markov chain; see for exampleGillman, Kejak, and Pakoš(2012).

amounts to specifying the transition probability matrix

P(h) =



1−λ12h λ12h λ21h 1−λ21h





in terms of positive transition probability intensities λij for i 6= j. ⁶ Note that the duration of high-growth and low-growth periods is exponentially distributed as follows

duration of high growth|St= 1 ∼ Exp (λ12), duration of low growth|S_t= 2 ∼ Exp (λ21),

therefore the hazard rate of a transition is constant and the distribution displays memoryless property.

Our setting nests this canonical model of the business-cycle fluctuations in div- idend growth rates. Our natural extension to a three-state Markov chain with the state space S defined above features one high-growth state µ^e_expansion for e∈ E and two low-growth state of equal growth µ^e_recession =µ^elost decade for e ∈ E. We let the state 1 correspond to the business cycle expansion and let the states 2 and 3 cor- respond to two different types of slowdowns in economic activity. The first type is fairly common but also fairly brief; it has all the characteristics of the business-cycle recession. The second type is much rarer and occurs on average once a century.

Moreover, it is also very persistent as in our calibration it lasts on average ten years.

Following the experience of Japan in the 1990s, we refer to this state as the “lost decade”. A statistical way to think of our setting is to picture a nature that tosses a biased coin the outcome of which decides whether the business-cycle recession is

6The intensity represents the instantaneous risk of moving from state i to state j, λij = limh↓0Prob (St+h=j|St=i)/h. The intensities form a matrix whose rows sum to zero so that the diagonal entries are defined byλii=−P

j6=iλij. See for exampleKarlin and Taylor(1975) for a thorough introduction to the theory of continuous-time Markov chain.

going to be the short type or the long type.

The setting of our three-state model is observationally equivalent to the well- accepted two-state models for the dividend growth rates. The key difference is that the hazard rate of the low-growth state is stochastic rather than constant. In other words, the nature tosses a biased coin and its outcome decides the average duration of the slowdown in economic activity.

With these ideas in mind, we specify the transition probability matrix as

P(h) =









1−(λ12+λ13) h λ12h λ13h λ21h 1−λ21h 0 λ31h 0 1−λ31h







.

The transition intensities λ12, λ13, λ21, λ31 are all strictly positive. In addition, although we rule out “instantaneous” transitions between the two types of slow- downs in economic activity, these states nonetheless do “communicate” because Prob (S_t+T =j|S_t=i) is strictly positive for any positive intervalT and anyi, j∈ S.

We believe that the three-state Markov chain is by far the most convenient math- ematical device to introduce a tractable model of the business fluctuations in which the economy from time to time experiences low-growth periods of dramatically longer duration than the typical NBER downturn. This result cannot be obtained with two- state model calibrated to NBER recessions that features constant transition proba- bility matrix. The reason is that the duration of the states in the Markov chain are exponentially distributed but with a relatively small mean. And although particular realizations indeed differ, the differences are insufficient to generate a long recession.

For example, when the average duration of the downturn is a year, lost decades tend to occur about every 22,000th recession which is very rare. In contrast, our setting with hidden persistence is different because the distribution of the downturn dura-

tion is a time-varying mixture of exponentials and that implies heavier tails of the duration distribution.

2.3 Signal-Extraction Problem

We assume that the Markov state of the economy S_t is hidden from the investor.

However, its valueSt=idetermines the mean dividend growth rateE( d lnD_t^e|St=i) = µ^e_idtfor eache∈ E over the interval(t, t+ dt). The sum of the mean growth rate and the idiosyncratic shocksdZ_t^ethat continuously hit the economy determines the real- ized dividend growth rate d lnD_t^e. The idiosyncratic shocks mask the mean growth rate and thus the true hidden state. As a result, realized dividends are but a noisy signal of the underlying state of the economy. The investor’s inference problem is to optimally extract the current hidden state from the history of the dividend rate sig- nalsI_t=

D_τ^u, D^l_τ

for τ ≤t . The outcome of the inference is a discrete posterior distribution π_t= (π1,t, π2,t, π3,t)^′ with

πi,t = Prob (St=i| It). (2.4)

Furthermore, the dividend growth rates in equations (2.2) and (2.3) may be decom- posed into the predictable partE( d logD^e_t| It)and the unpredictable partd logD_t^e− E( d logD_t^e| It). The predictable part is related to the posterior distribution as

E( d logD_t^e| I_t) =E µ^e_StI_t

dt= X

i∈S

µ^e_iπ_i,t

!

dt=µe^e(π_t) dt.

The unpredictable part is the instantaneous forecast error. We normalize it to have variance one and denote it by tildes:

dZe_t^e = 1

σ^e (d logD_t^e−E( d logD_t^e| I_t)).

The decomposition into predictable and unpredictable parts implies that the realized dividend growth rate may be written as

d logD_t^e = µe^e(π_t) dt+σ^edZe_t^e for e∈ E. (2.5)

In the language of stochastic calculus, the process Zet =

Ze_t^u,Ze_t^l

is known as the innovation process. Liptser and Shiryaev (1977) show that the innovation process is a standard bivariate Brownian motion with respect to the investors filtration I = {It:t >0}. In the language of economics, the unpredictable part dZe_t^e is a source of dividend news regarding the hidden stateSt because

dZe_t^e = µ^e_S

t−eµ^e(π_t) σ^e

dt+ dZ_t^e.

Nonzero realizations of dZe_t^e may come from the I.I.D. dividend shock dZ_t^e but also from the hidden transitions of the Markov state S_t. The term

µ^e_S

t −µe^e(π_t) /σ^e measures the regime change in units of the volatility of the noise and may be thought of as the signal-to-noise ratio. Large signal-to-noise ratio tends to be particularly informative about the regime change.

When we recursively apply the Bayes rule to eq. (2.4), we obtain the filtering equations ofWonham (1964):

dπ_t = η(π_t) dt+X

e∈E

ν^e(π_t) dZe_t^e, (2.6)

where the exact functional form of the drift η =η(π) and volatility ν^e = ν^e(π) is presented in Appendix A in Lemma 1.

2.4 Precision of Consumption and Dividend Forecasts and Eco- nomic Uncertainty

Economic uncertainty is measured by the precision of the long-run consumption and dividend forecasts; for example, times of heightened uncertainty are characterized by increased variance of the forecast errors. Such economic uncertainty is priced using the Epstein-Zin preferences that are configured so that an early resolution of uncertainty is preferred. From an asset pricing perspective, persistent variation in the forecast precision of the long-run consumption and dividend growth rates induces a corresponding variation in the uncertainty (or timing) discounts in asset prices.

We decompose the forecast error variance of the long-run dividend growth rates into the sum of two terms. The first term is the forecast error variance of the mean long-run dividend growth rates. The second term is the forecast error variance of the long-run idiosyncratic shocks. This decomposition is relevant because it identifies two sources of variation in the forecast precision and thus allows us to compare hidden Markov model for consumption and dividends to the canonical long-run risk model of Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2010). We find that when consumption and dividends are subject to hidden Markov shifts, the forecast error variance of the mean long-run growth growth fluctuates in response to the perceived economic uncertainty. In contrast, the forecast error variance of the mean long-run growth is constant in the Bansal-Yaron economies which means that unless the variance of the long-run idiosyncratic shocks changes the forecasts of the long-run growth display constant precision. Second, our model of the consumption and dividend growth rate implies that variance of the long-run idiosyncratic shocks is constant whereas it follows an exogenous mean-reverting process in the Bansal-Yaron economies.

In order to calculate the variance decomposition of the long-run forecast errors,

we first calculate the long-run forecasts and the forecast error variance conditional on the hidden Markov state. We then condition down to the investor’s coarser infor- mation sets. Finally, we compare the hidden Markov model of expected consumption and dividends to the corresponding AR(1) model from Bansal-Yaron economies.

2.4.1 Forecasts Conditional on the Hidden Markov State

The instantaneous forecasts for the growth rates are E( d logD_t^e|St=i) = µ^e_idt for each e ∈ E and i ∈ S and the corresponding forecast errors are d logD^e_t −

E( d logD_t^e| It). The instantaneous homoscedasticity of the forecast errorsvart[d logD^e_t−E( d logD^e_t| It)] = (σ^e)²dtis a consequence of the Girsanov Theorem. In contrast, the long-run forecasts

of the mean dividend growth ratesg^e_T|i=E

´t+T

t d logD^e_τ

St=i

=E

´t+T t µ^e_τdτ

St=i can be calculated in semi-closed form as described in Appendix D.4. Note that be-

cause the chain may transition a number of times during the long forecast period (t, t+T), the approximation forg_T^e_|i asµ^e_i×T is imprecise. In addition, the instan- taneous forecastsµ^e_i as well as the long-run onesg_T^e_|i are both conditional onSt and therefore evolve as dependent Markov chains as well.

2.4.2 Forecasts Conditional on the Investor’s Information

The investor’s information setsI_tdo not contain the hidden stateS_t. Thus, the op- timal forecasts with respect toI_tmust be calculated asg^e_T|π =P

i∈Sg^e_T|iπ_i.Further- more, the forecast error variancev_T^e_|π = var

´t+T

t d logD^e_τIt

may be decomposed into a sum of the forecast error variance of the mean long-run growth rate,

var

ˆ t+T

t

µ^e_τdτ It

= X

i∈S

g_T^e_|i2

πi− X

i∈S

g_T^e_|iπi

!2

,

and the variance of the long-run idiosyncratic shocks,

var

ˆ t+T

t

σ^edZτ

It

= (σ^e)²T.

Both the mean long-run growth forecast as well as its variance necessarily depend on the posterior distributionπ= (π1, π2, π3)^′ and thus change in response to the degree of the perceived economic uncertainty. The variance of the long-run idiosyncratic shocks equals (σ^e)²T and is thus constant for a given forecast horizon.

2.4.3 Comparison to Bansal-Yaron Economies

Our cash-flow model with hidden Markov shifts differs from the canonical long- run risk models of Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2010) in several important dimensions. First, the variance of the mean long-run growth var

´t+T t µ^e_τdτ

It

is constant in the Bansal-Yaron economies because the shocks to the expected consumption growth are by assumption homoscedastic. In contrast, when expected consumption growth follows hidden Markov chain, shocks to the instantaneous expected consumption growth µe^e(π) are necessarily heteroscedastic as proved by Veronesi (1999, Proposition 6). Second, the variance of the long-run idiosyncratic consumption shocks in Bansal-Yaron economies is specified exogenously but is constant in our setting.

The key implication is that the setting with hidden regime shifts does not need the assumption of exogenous consumption volatility changes in order to generate fluctuating precision in the long-run growth forecasts. In fact, even with constant volatility of the idiosyncratic consumption innovations var

´t+T

t σ^edZ_τI_t , the forecast error variance of the long-run growth changes endogenously in response to the variation in economic uncertainty as measured by the dispersion of the posterior distribution π.

2.5 Consumption-Portfolio Problem

The recursive structure of the investor’s consumption-portfolio problem leads to the Hamilton-Jacobi-Bellman equation

0 = max

{c, ωafora∈A}

u(c, v) + 1

dtEt(dv)

(2.7)

subject to the dynamic budget constraint

dwt = wtdr_t^w−ctdt, (2.8)

where w_t is the investor’s financial wealth and the return on the wealth portfolio dr^w_t is defined by

dr_t^w = X

a∈A

ωa,tdr_t^a+ 1−X

a∈A

ωa,t

! r_t^fdt,

where r_t^f is the short-term riskless rate andωa,tis the portfolio share of the asset a.

The first-order condition w.r.t. to the consumption rate c yields the standard condition that the marginal utility of consumption u_c equals the marginal utility of wealth v_w. The first-order conditions w.r.t. portfolio weights ω_a for each a ∈ A state that the risk premiums equal the covariance of the realized returns with the negative of the marginal utility growth; see equation (2.15) in the next section.

In order to make analytical progress, we conjecture that the value function is separable across the beliefs(π1, π2) and the financial wealth w,

v(w, π1, π2) = h(π1, π2)w^1−γ

1−γ, (2.9)

where π3 is given implicitly as 1−π1−π2. The first-order condition with respect

to the consumption rate implies that the function h = h(π1, π2) depends on the equilibrium price-dividend ratio of the Lucas tree,

h(π1, π2) =δ

1−γ 1−1 ψ

P^u D^u

^1−γ_ψ−1 .

Equilibrium in the goods market stipulates that the aggregate demand for the con- sumption good equals the aggregate supply, ct = D_t^u. Equilibrium in the capital market stipulates that the aggregate demands for the unlevered and levered equity and real bonds equal their fixed supply, ω^u_t = 1, ω_t^l = 0 and ω^b_t = 0 for any given t >0.

2.6 Marginal Utility and Asset Prices

When the innovations in the dividends are Brownian motions, the equilibrium asset prices necessarily follow diffusion as shown inHuang (1987). In our case, the return dynamics for eacha∈ A has the following structure:

dr_t^a = η_t^adt+X

e∈E

σ^e_a,tdZe_t^e, (2.10)

where

dr_t^a = dP_t^a+D_t^adt P_t^a .

The expected return η_t^a and the return volatilities σ^u_a,t, σ^l_a,t

are determined by market clearing in general equilibrium. In fact, the first-order conditions with respect to the portfolio weights ωa state

Et(dr_t^a)−r^fdt = −covt

dr^a_t,dMt

Mt

(2.11)

whereMt= exp

´t

0uv(cτ, vτ) dτ

uc(ct, vt)is the marginal utility of the representa- tive investor. Itô lemma says that the growth rate of the marginal utilityM_tevolves as

dMt

Mt

= −r^f(πt) dt−X

e∈E

Λ^e(πt) dZe_t^e. (2.12)

As usual, the negative of the expected marginal utility growth rate equals the short riskless rater^f. Furthermore, the volatilityΛ^e(π_t)is the so-called risk-price function and it measures the contribution of a marginal exposure to the shockdZe_t^e fore∈ E to the total expected return on any asseta∈ A; see eq. (2.15). The functional forms for Λ^u andΛ^l

Λ^u(π1, π2) = γσ^u+ 1− 1−γ 1−_ψ¹

! ₂ X

i=1

ν_i^u Φ^u_i

Φ^u

, (2.13)

Λ^l(π1, π2) = 1− 1−γ 1− _ψ¹

! ₂ X

i=1

ν_i^l Φ^u_i

Φ^u

(2.14)

reveal that the risk-price functions depend on the price-dividend ratio of the Lucas tree Φ^u =P^u/D^u as well as the respective partial derivatives Φ^u_i =∂Φ^u/∂πi. The interpretation of the constant term γσ^u inΛ^u is as the Lucas-Breeden component.

In fact, eq. (2.15) implies that part of the risk premiums on assets a∈ A derives from the covariance with the consumption growth times the risk aversion coefficient.

We refer to it as the “short-run risk” and it plays a marginal role in accounting for asset pricing phenomena in our model. The remaining component(s) in Λ^e for e ∈ E reflect the equilibrium compensation for the late resolution of the dividend uncertainty associated with investing into long-lived assets. Such “timing” premium is an equilibrium outcome of the investor’s hedging demand for the asset in response to the fluctuations in their own uncertainty; seeMerton(1973) andVeronesi(1999).

Expected utility of von Neuman and Morgenstern is nested as a special case γ = 1/ψ with zero timing risk premiums as the independence axiom implies that agents exhibit neither preference nor dislike for the timing of resolution of the dividends uncertainty and thereforeΛ^u=γσ^u and Λ^l = 0.

Using the definitions of the risk prices (2.13) and (2.14) allows us to rewrite the first-order conditions with respect to the portfolio weights ω^a for a∈ Aas

η^a−r^f = X

e∈E

Λ^eσ^e_a (2.15)

by direct substitution from equations (2.10) and (2.12). Finally, although asset prices P_t^a for a∈ A \ {b} have unit roots and tend to drift upwards, the respective price-dividend ratios

Φ^a(π1t, π2t) ≡ P_t^a

D_t^a (2.16)

are only functions of the posterior distribution that evolves as a stationary stochastic process. Proposition 1 in Appendix B then allows us to express these as well as the zero-coupon real bond priceP_t^b≡P^b(π1t, π2t, t;T)as solutions of certain differential Fichera boundary value problems.

3 Data Description

We measure nominal consumption as the sum of nominal personal consumption ex- penditures on nondurable goods (PCND) plus services (PCESV). The data are quar- terly from 1952:I to 2011:IV and are available from the Federal Reserve Economic Data at the Federal Reserve Bank of St. Louis. We convert the nominal consump- tion series to real per-capita basis by dividing by the population (POP) and deflating

by the end-of-quarter consumer price index (CPIAUCSL). Furthermore, we update Bansal, Gallant, and Tauchen (2007) by constructing the asset return and dividend series from 1952:I to 2011:IV. We measure the return on the market portfolio as the value-weighted return, and the dividend as the sum of total dividends, on all the NYSE and the AMEX common stocks. The data are monthly from 1952:1 to 2011:12 and are available from the Center for Research in Security Prices (CRSP) at the Uni- versity of Chicago. We start by constructing the monthly stock market valuation series as the month-end value of the nominal capitalizations of the NYSE and the AMEX. We compute the monthly dividend series on the NYSE and AMEX as the difference between the nominal value weighted total return and the value-weighted capital gain. We apply this dividend yield to the preceding month’s market capital- ization to obtain an implied monthly nominal dividend series. We remove the large seasonal component in monthly nominal dividend series by using the X-12-ARIMA Seasonal Adjustment Program, available from the U.S. Census Bureau. We compute the quarterly real geometric return and dividend growth rate series as the monthly nominal continuously-compounded return and dividend on the NYSE and AMEX, cumulated over the quarter to form nominal quarterly series and deflated by the con- sumer price index (CPIAUCSL). The real dividend series is in addition converted to real per-capita basis by dividing by the population (POP). The real interest rate is the one-month nominal Treasury bill log yield that is aggregated to quarterly values and deflated by the consumer price index (CPIAUCSL).

Table 1 presents the summary statistics. The annualized quarterly mean log consumption growth rate is about 1.87 percent with the standard error of about 0.10 whereas the annualized mean log dividend growth rate is about 2.06 percent but higher standard error of about 0.36. We cannot reject the null hypothesis for the equality of the means as the bootstrap confidence interval at 5 percent significance

level for the series difference comes out (-0.37, 0.28). Furthermore, the annualized consumption volatility of 1.26 percent is almost an order of magnitude smaller than the dividend volatility of about 10.38. In addition, there are economically negligible first-order autocorrelations in consumption and dividend growth rates. As regards the asset market data, the mean short riskless interest rate is about 1.01 percent per year with volatility of about 1.34 percent whereas the annual equity premium is about 5.51 percent. We easily reject the null hypothesis of zero equity premium.

Furthermore, the annual return volatility comes out about 16.55 percent and the annualized equity premium autocorrelation is insignificant. In addition, the average annual Sharpe ratio in our sample is about 0.33. Finally, the mean dividend yield is about 3.19 percent per year with a large annual volatility of 34 percent and annual persistence of 0.82 with the standard error of 0.09.

4 Inference

We estimate the unknown parameters for unlevered and levered dividends in the bi- variate continuous-time Hidden Markov model in equations (2.2)–(2.3) by maximum likelihood from the realized quarterly consumption and dividend series. Data used as well as the parameter estimates that we report in Table2are expressed per quarter.

In addition, Figure1 plots the realized consumption and dividend log growth rates, the realized dividend to consumption ratio and the estimated beliefs for the sample period 1952:I – 2011:IV.

In our approach we take into account the fact that the outcomes are discretely observed by calculating the transition probabilities as the elements of the matrix exponential of the transition intensity matrix

λ=









−(λ12+λ13) λ12 λ13

λ21 −λ21 0

λ31 0 −λ31







.

Lost decades are arguably relatively rare events and we cannot hope to identify all these parameters based on 60×4 = 240 quarters of data that we have. We thus impose the arguably plausible restriction that lost decades last on average a decade and thus the inverse transition intensity λ⁻₃₃¹ equals 40 quarters. Furthermore, we posit that lost decades, such as the Great Depression, tend to occur only once a century and thus restrict the stationary probability of the chain asπlost decade= 0.1.

Next, the fact that the data are sampled at quarterly frequency makes it arguably difficult to detect within-quarter transitions. We thus allow only for end-of-quarter transitions and calculate the mean quarterly growth rate E

´t+1

t d lnD_τ^eSt=i as E

´t+1 t µ^e_τdτ

St=i

= µ^e_i × ´t+1

t dτ = µ^e_i where we use the fact that Z_t^e is a zero-mean martingale that is statistically independent of the Markov chain S_t. In addition, we are able to calculate the conditional quarterly variance as var

´t+1

t d lnD^e_τS_t=i

asvar

´t+1

t σ^edZ_τ^e

= (σ^e)²×´t+1

t d [Z^e]_τ = (σ^e)² for all e∈ E where we again invoke the independence between Z_t^e and S_t. The restriction of only end-of-quarter transitions moreover allows us to easily construct the outcome probability Prob D^u_t, D_t^lSt=i

as the bivariate Gaussian distribution with mean µ^u_i, µ^l_i′

and variance-covariance matrixdiagn

(σ^u)², σ^l2o

. Note that we restrict the growth ratesµ^e2 andµ^e3 in the two slowdowns in economic activity to be exactly equal which is our way of modeling hidden persistence. ⁷ In addition, we restrict the dividend volatilities σ^u and σ^l to be constant across the hidden regimes which is exactly consistent with our parsimonious specification in equations (2.2)–(2.3) when

7The reason for that was discussed before; equal growth rates in the downturn states effectively introduce stochastic average downturn duration into the well-accepted two-state Markov model of business-cycles.

regime shifts may occur only at the end of the quarter.

Overall, the vector of unknown parameters isp= λ11, λ12, λ13, λ22, µ^u₁, µ^u₂, σ^u, µ^l₁, µ^l₂, σ^l′

. The construction of the likelihood functionL(p| IT)followsHamilton(1989). Next,

we maximize the likelihood by initiating the Markov chain in its stationary distribu- tion. We then invoke several global optimization algorithms over 10×1 parameter vectorpwhere we must impose upper and lower bounds on all the estimated param- eters. We subsequently polish the optimum so found to greater accuracy using local derivative-free algorithm. The achieved log-likelihood islogL= 1256.13. Table2re- ports the point estimates as well as the standard errors for the transition intensities, and consumption and dividend growth rates and volatilities.

4.1 Transition Probability Intensities

The average duration of the high-growth expansion period comes out aboutˆλ⁻₁₁¹≈ 23.8quarters or about 6 years whereas the average duration of the brief low-growth recession is aboutλˆ⁻₂₂¹≈4.2quarter, or about a year. Furthermore, the probability q12= _λ ^λ12

12+λ13 of a transition from expansion to short-lived recession, conditional on the transition occurring, comes out qˆ12 ≈0.92 and thus about every 12th–13th re- cession tends to be a protracted lost decade. Moreover, the fact that the stationary distribution of the chain is estimated to be π= (0.73,0.17,0.10)implies that a par- ticular century tends to experience about 73 years of good-times that are interrupted by about 17 years of the brief business-cycle recessions, each of the average duration of about a year, where the remaining 100−73−17×1 = 10years are spent in the period of the long anemic economic growth (i.e., lost decade).

4.2 Instantaneous Forecasts

The instantaneous rates of consumption growth in the high-growth state and low- growth state are estimated to be quite conservative; consumption tends to grow on average at the annualized rates of about 0.592%×4 = 2.37% and −0.267%×4 =

−1.07%, respectively. Conservative estimates for the growth rates µˆ^u₁ and µˆ^u₂ = ˆµ^u₃ decrease the consumption risk in the economy which tends to worsen the asset- pricing implications. Furthermore, we estimate the consumption volatility to be ˆ

σ^u = 0.543%×2 = 1.09%p.a. The estimates for dividends are conservative as well.

The high-growth rate is estimated to be µˆ^l1 = 0.764%×4 = 3.06% whereas the growth rate in the recession period isµˆ^l2 =−7.55%. Conservative estimates for the growth ratesµˆ^l1 andµˆ^l2 = ˆµ^l3 decrease the dividend risk in the economy which tends to worsen the asset-pricing implications for the aggregate stock market. Finally, we estimate the dividend volatility to beσˆ^l = 5.08×2 = 10.1%p.a.

Compared to the latest asset-pricing literature, the calibration of the two-state HMM model inJu and Miao(2012, Table II) specifies the consumption growth rate in the low-growth period to be -6.79% per year. Our consumption growth rate of -1.07% per year is arguably more conservative. In addition, their calibration of the dividend growth rate in the low-growth period is −6.79%×3 =−20.37% per year, where the number 3 reflects the dividend leverage, whereas our dividend growth rate of −7.55%per year is again arguably more conservative.

4.3 Long-Run Forecasts

The annual consumption growth rate forecasts are g_T^u_=4|i=1 = 2.16% in expansions whereasg_T^u_=4|i=2= 0.09%in downturns andg_T^u_=4|i=3 =−0.9%in lost decades where time is measured in quarters. The reason why the consumption growth is forecast to grow at positive rate of 0.09% during downturn is simple; the average duration of

a downturn is one year and hence a transition to the expansion state is very likely.

In addition, the decade-long forecasts are also quite relevant. In particular, consumption growth is forecast to grow g_T^u_=40|i=1 = 1.81% expansions, g_T^u_=40|i=2 = 1.51%in downturns and g_T^u_=40|i=3 = 0.01% in lost decades. In fact, the lost decade s = 3 may be thought of as a protracted, decade-long, period of anemic growth during which the consumption level is forecast to remain the same. Note that the estimation procedure does not impose any constraints on the consumption growth rates themselves; it restricts only the average duration λ⁻₃₃¹ = 10 years and the relative frequency in century-long series (i.e., πlost decade = 0.1). To that extent, the fact that the decade-long consumption-growth forecast comes out zero is dictated by the realized consumption and dividends series but nonetheless it is consistent with our interpretation of the long recession as the lost decade.

As regards dividends, the decade-long forecasts come out g^l_T=40|i=1 = 1.32% in expansions,g^l_T=40|i=2 = 0.42%in downturns andg_T^l_=40|i=3=−4.23%in lost decades.

For example, the cumulative drop in dividends over the whole 10-year duration of the rare lost decade is forecast to be −4.23%×10 = −42.30% which is quite a plausible number. For example, the calibration of the two-state HMM model in Ju and Miao (2012, Table II) implies that the cumulative drop in the dividends during their down-state is about −6.79%×3×2 =−40.74%, where the number 3 reflects the dividend leverage and the number 2 is the average duration of the down state.

However, our model predicts dividends to fall by about -42% once a century whereas Ju and Miao predict the dividends to fall by about -40% during each business-cycle downturn. In fact, our model predicts that the mean cumulative decline in realized dividends over the business-cycle recessions is about −3.98%×1 =−3.98%. This quantitative comparison shows that much less dividend risk, measured in terms of the difference in the hidden growth rates µ^u₁ −µ^u₂ and µ^l₁−µ^l₂, is needed when the

growth persistence itself is subject to change, as opposed to the common two-state models of asset prices that feature constant persistence.

5 Asset-Pricing Implications

We first assess the performance of the model in terms of the unconditional moments for unlevered and levered equity as well as the short- and long-term yields on real zero-coupon bonds. Because we do not have closed-form solutions for the moments, we use Monte Carlo methods to integrate conditional moments with respect to the stationary density for the state variables – the posterior distribution. In addition, we use Hamilton (1989) filter to estimate the beliefs πbi,t = Prob (d St=i| It) and then assign asset prices to such sequence of estimated beliefs. We plot the so- calculated asset prices in Figure 3. Second, we discuss the properties of the price- consumption and price-dividend ratios and explain their convexity as the outcome of the investor’s hedging demands against his own uncertainty. Third, we calculate the long-horizon equity premium and equity volatility by proper time aggregation which leads to a Fichera boundary value problem to be solved numerically. We explain how the variation in economic uncertainty gives rise to endogenous changes in risk premiums and volatility. Fourth, we evaluate the serial correlation in the dividend growth rates and excess equity returns. We calculate variance ratios and find that in contrast to the dividend growth rates excess returns display pronounced mean reversion with variance ratios well below one. The results of the predictive regressions furthermore indicate that long-run dividend growth rates are close to unpredictable whereas the results for the excess returns display increasing slope coefficients and R²s, consistent with the empirical finance literature. Fifth, we calculate the average excess returns, volatility and Sharpe ratios across different phases of the business cycle. Our model is the first long-run risk model to account for the decreasing pattern

of excess returns, volatility and Sharpe ratios over the phases of the expansion and the increasing pattern of the excess returns, volatility and Sharpe ratio over the recession. Our Monte Carlo results produce average risk premiums and Sharpe ratio over the expansion and recessions that are surprisingly close to the point estimates reported in Lustig and Verdelhan(2013, Table 2).

5.1 Unconditional Moments

Table 4 in Panel A reports the pricing moments from the Monte Carlo simulation to be compared to Table 1. First, the riskless interest rate measured as the yield- to-maturity on a real zero-coupon bond maturing in 3 months comes out about 1.20 percent per year with annual volatility of about 0.59 percent whereas the eq- uity premium on the levered long-lived asset is about 5.89 percent per year with a large volatilityσ

E_t

R_t+1−R^f_t

of about 3.61 percent. The average conditional volatility of returnsE

σ_t

R_t+1−R^f_t

comes out about 15.88 percent and varies a lot across the cycle; its standard deviationσ

σ_t

R_t+1−R^f_t

is about 2.32 percent.

Note that the cautionary observation in Abel (1999) that one may be accidentally accounting for the large equity premium with a large term premium does not apply as the yield-to-maturity on a long-term zero-coupon bond− ₃₀¹

E logP_t,^b30

comes out about 0.08 percent per year. The average dividend-yield E D_t^l/P_t^l

is large, about 4.26 percent with the volatility σ logD_t^l−logP_t^l

of 11.96 percent, about a half in comparison to the sample estimates in the literature, which is large especially if we take into account the fact that the economy grows at either high or low rate.

The dividend yield is also quite persistent, its first-order auto-correlation is about 0.80 when annualized. When the estimated beliefsbπtare inserted into the model, the performance reported in Table 4 (Panel B) partially weakens but many important moments remain almost intact. The equity premium comes about more than 5.00

percent per year, for example. We conclude that the model does a surprisingly good job of accounting for the average returns on financial assets.

5.2 Price-Consumption and Price-Dividend Ratios

Figure2(top left) plots the price-consumption ratioΦ^u as a function of the posterior probabilities (π1, π2) and Appendix describes the numerical approach used. The graph lies in the plausible interval 96 and 106 and has the additional two properties.

First, the price-consumption ratio is a strictly increasing function on the whole simplex domain. This result is directly implied by our ordering of the states as µ^e1 > µ^e2 = µ^e3 for e ∈ E. Second, the price-consumption ratio is also a convex function on the whole domain. This result, first analyzed by Veronesi (1999) with time-additive exponential utility, has the following economic intuition. The high- growth state tends to last long enough for a high confidence about the regime to develop. As the economy is constantly hit by Gaussian shocks, consumption news tend to positive as well as negative with equal probability. The positive news are however fairly uninformative; there is no higher growth state to shift to and therefore the news only reinforce the already high confidence about the up state. Negative news, though, are confusing. They may originate not only from a hidden regime shift but also from a disappointing idiosyncratic innovation in consumption growth without any change of the regime at all. It is difficult to disentangle the correct source of the news but the Bayes rule in fact suggest to partially adjust the beliefsπ1 as well asπ2+π3 toward 1/2 in response to negative consumption growth innovation. The end result is that the expected consumption growth eµ^u = eµ^u(π) falls and so does the value of the Lucas tree. What is more, such rise in the economic uncertainty tends to lengthen the time until the uncertainty partially resolves in terms of the posterior odds. That is particularly disliked by the Epstein-Zin households and